The Milne problem: A study of the mass dependence

The Milne problem is considered with the Boltzmann equation without approximation to the collision operator. The general solution is given as the sum of spatial transient and asymptotic functions. Solutions of the Boltzmann equation with a hard-sphere cross section are obtained by expansions in Burnett functions. Convergence of the distribution function and associated quantities, such as spatial eigenvalues, extrapolation length, diffusion coefficient, and density profile is studied. It is found that convergence can be achieved with the expansion in Burnett functions. In particular, the importance of retaining a large number of polynomials in angle is demonstrated. The dependence of the solution and associated quantities on the mass ratio $\gamma =\frac{m_1}{m}$ is examined, where $m_1$ and $m$ are the mass of medium and test particles, respectively. Wherever possible, comparison with the results of previous researchers is given.